Wave-Point Topology and Topological Chemistry: From Combinatorial Geometrization to Non-Local Molecular Modeling

This submission presents a unified, two-part foundational architecture that bridges pure discrete differential geometry with predictive mathematical chemistry, establishing the paradigm of "Wave-Point Topology." Part 1 ("Seed Crystals: Atomic Crystallization of Polyhedra via Convective-Deformation Flows") provides a purely combinatorial and dynamic resolution to Thurston's Geometrization Conjecture and the 3-dimensional Poincaré Conjecture. By mapping a three-dimensional manifold onto a discrete polyhedral triangulation network, geometric deformations are modeled as a non-linear coupled flow driven by wave conductivity (phase connectivity) and wave amplitude (energy density). The work rigorously proves that topological stress forces bottlenecks to undergo a double-exponential metric collapse, smoothly splitting the manifold into rigid geometric crystals or a state of absolute structural zero characterized by three irreducible ultimate phases. Part 2 ("Topological Chemistry: Modeling Non-Local Molecular Forces and Crystalline Phase Transitions via Wave-Point Geometry") extends this geometric machinery into a coordinate-free framework for structural material design. By formalizing long-range linear fields—such as Coulombic interactions and intermolecular van der Waals forces—directly as dynamic non-local edge tensors within the network (gᵤᵥᶠᵒʳᶜᵉ), chemical physics are governed entirely by the spectrum of a gauge-fixed Pinned Laplacian matrix. This algebraic approach successfully predicts three unprecedented, topologically constrained molecular structures (the Fractional Hyperbolic Carbon Schwarzite, the Topological Coaxial Molecular Wire, and Spontaneous Homotopic Spherical Cages) without relying on traditional spatial coordinate mechanics or heavy quantum computations. Together, these works demonstrate that both the macro-scale structure of space and the micro-scale geometry of matter are emergent phenomena governed by the discrete, eternal harmony of wave-points and algebraic graph kernels.